The following is an argument against the compatibility of divine foreknowledge and libertarian free will:
α: ☐p → Np, where Np means, "there is nothing that we can do to change p."
β: [(Np & N(p → q)) → Nq]
Let S(t) be the state of the world at t (where t is some time in the distant past), and S(t) includes God's knowing that Q, where Q = "I will eat cheese" Then:
1. ☐(S(t) → Q)
By α,
2. N(S(t) → Q)
3. NS(t) [prem, I cannot change the past]
4. NS(t) & N(S(t) → Q)
By β,
5. NQ
Conclusion: no matter what I do, I will eat cheese. Therefore, I am not free